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A239116 Number of ballot sequences of length n with exactly 5 fixed points.

%I A239116 #14 Feb 06 2017 18:03:54
%S A239116 0,0,0,0,0,1,1,3,9,29,99,356,1343,5279,21584,91324,399456,1799568,
%T A239116 8343404,39702144,193768604,967992476,4946617328,25817913584,
%U A239116 137549830384,747137750064,4135349698416,23301072909248,133591802704944,778722128953904,4613070010373504
%N A239116 Number of ballot sequences of length n with exactly 5 fixed points.
%C A239116 The fixed points are in the first 5 positions.
%C A239116 Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, ..., 5, but not 6.  An alternate definition uses the first row.
%H A239116 Joerg Arndt and Alois P. Heinz, <a href="/A239116/b239116.txt">Table of n, a(n) for n = 0..800</a>
%H A239116 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A239116 See Maple program.
%F A239116 Recurrence (for n>=7): (n-6)*(n^5 - 21*n^4 + 157*n^3 - 699*n^2 + 3298*n - 13680)*a(n) = (n^6 - 27*n^5 + 235*n^4 - 537*n^3 - 1964*n^2 - 2316*n + 54720)*a(n-1) + (n-7)*(n-5)*(n^5 - 16*n^4 + 83*n^3 - 344*n^2 + 2292*n - 10944)*a(n-2). - _Vaclav Kotesovec_, Mar 11 2014
%F A239116 a(n) ~ sqrt(2)/288 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - _Vaclav Kotesovec_, Mar 11 2014
%e A239116 a(5) = 1: [1,2,3,4,5].
%e A239116 a(6) = 1: [1,2,3,4,5,1].
%e A239116 a(7) = 3: [1,2,3,4,5,1,1], [1,2,3,4,5,1,2], [1,2,3,4,5,1,6].
%e A239116 a(8) = 9: [1,2,3,4,5,1,1,1], [1,2,3,4,5,1,1,2], [1,2,3,4,5,1,1,6], [1,2,3,4,5,1,2,1], [1,2,3,4,5,1,2,3], [1,2,3,4,5,1,2,6], [1,2,3,4,5,1,6,1], [1,2,3,4,5,1,6,2], [1,2,3,4,5,1,6,7].
%p A239116 a:= proc(n) option remember; `if`(n<6, [0$5, 1][n+1],
%p A239116       ((952098*n^4 -28186656*n^3 +321186690*n^2 -1739275812*n
%p A239116        +3721544280)*a(n-1) +(n-7)*(451397*n^4 -9536389*n^3
%p A239116        +64448100*n^2 -229993164*n +534842280)*a(n-2)
%p A239116        -(n-7)*(n-8)*(500701*n^3 -9933473*n^2 +95681400*n
%p A239116        -319342500)*a(n-3))/
%p A239116        ((n-5)*(451397*n^3-9487085*n^2+55580742*n-95239584)))
%p A239116     end:
%p A239116 seq(a(n), n=0..40);
%t A239116 b[n_, l_List] := b[n, l] = If[n <= 0, 1, b[n - 1, Append[l, 1]] + Sum[If[i == 1 || l[[i - 1]] > l[[i]], b[n - 1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; a[n_] := If[n == 5, 1, b[n - 6, {2, 1, 1, 1, 1}]]; a[n_ /; n < 5] = 0; Table[ Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* _Jean-François Alcover_, Feb 06 2015, after Maple *)
%Y A239116 Column k=5 of A238802.
%K A239116 nonn,easy
%O A239116 0,8
%A A239116 _Joerg Arndt_ and _Alois P. Heinz_, Mar 10 2014